Real numbers are of obvious importance in physics, and electrodynamics
is no exception. Measured or counted quantities are almost invariably
described in terms of real numbers or their embedded cousins, the
integers. Their virtue in physics comes from from the fact that they
form a (mathematical) field4.1 that is,
they support the mathematical operations of addition, subtraction,
multiplication and division, and it empirically turns out that physical
laws turn out to be describable in terms of algebraic forms based on (at
least) real numbers. Real numbers form a group under ordinary
multiplication and, because multiplication is associative and each
element possesses a unique inverse, they form a division
algebra4.2

A division algebra is one where any element other than zero can be
divided into any other element to produce a unique element. This
property of real numbers is extremely important - indeed it is the
property that makes it possible to use algebra per se to solve for many
physical quantities from relations expressed in terms of products and
sums. The operational steps:

(4.1)

are so pervasively implicit in our algebraic operations because
they are all learned in terms of real numbers that we no longer even
think about them until we run into them in other contexts, for example
when are matrices, with at least being an invertible
matrix.

In any event real numbers are ideally suited for algebra because they
form a field, in some sense the archetypical field, whereby physical law
can be written down in terms of sums and products with measurable
quantities and physical parameters represented by real numbers. Other
fields (or rings) are often defined in terms of either subsets of the
real numbers or extensions of the real numbers, if only because when we
write a symbol for a real number in an algebraic computation we know
exactly what we can and cannot do with it.

Real numbers are the basis of real ``space'' and ``time'' in physics -
they are used to form an algebraic geometry wherein real numbers
are spatiotemporal coordinates. This use is somewhat presumptive
- spacetime cannot be probed at distances shorter than the Planck
length (
meters) - and may be quantized and
granular at that scale. Whatever this may or may not mean (close to
nothing, lacking a complete quantum theory of gravity) it makes no
meaningful difference as far as the applicability of e.g. calculus down
to that approximate length scale, and so our classical assumption
of smooth spacetime will be quite reasonable.

Are real numbers sufficient to describe physics, in particular
classical electrodynamics? The answer may in some sense be yes
(because classical measurable quantities are invariably real, as are
components of e.g. complex numbers) but as we will see, it will be far easier to work over a different field: complex numbers,
where we will often view real numbers as just the real part of a more
general complex number, the real line as just one line in a more general
complex plane. As we will see, there is a close relationship between
complex numbers and a two-dimensional Euclidean plane that permits us to
view certain aspects of the dynamics of the real number valued
measurable quantities of physics as the real projection of dynamics
taking place on the complex plane. Oscillatory phenomena in general are
often viewed in this way.